Related papers: On 3-Connected Planar Graphs with Unique Orientabl…

A graph is $1$-planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges…

We say that a graph is intrinsically knotted or completely 3-linked if every embedding of the graph into the 3-sphere contains a nontrivial knot or a 3-component link any of whose 2-component sublink is nonsplittable. We show that a graph…

It is known that the canonical double cover of any connected nonbipartite graph have an automorphism group of the form $H \rtimes \mathbb{Z}_2$, where $H$ is the set of automorphism which preserve bipartite parts. We construct connected…

We prove that every 3-connected planar graph on $n$ vertices contains an induced path on $\Omega(\log n)$ vertices, which is best possible and improves the best known lower bound by a multiplicative factor of $\log \log n$. We deduce that…

A balanced graph is a bipartite graph with no induced circuit of length 2 mod 4. These graphs arise in linear programming. We focus on graph-algebraic properties of balanced graphs to prove a complete classification of balanced Cayley…

A \emph{$k$--bisection} of a bridgeless cubic graph $G$ is a $2$--colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes have order…

A graph is $k$-planar if it can be drawn in the plane such that no edge is crossed more than $k$ times. While for $k=1$, optimal $1$-planar graphs, i.e., those with $n$ vertices and exactly $4n-8$ edges, have been completely characterized,…

For any pair of edges $e,f$ of a graph $G$, we say that {\em $e,f$ are $P_3$-connected in $G$} if there exists a sequence of edges $e=e_0,e_1,\ldots, e_k=f$ such that $e_i$ and $e_{i+1}$ are two edges of an induced $3$-vertex path in $G$…

In this paper, we introduce the notion of 2-boundary planar graphs. A graph is 2-boundary planar if it has an embedding in the plane so that all vertices lie on the boundary of at most two faces and no edges are crossed. A proper coloring…

A path cover is a decomposition of the edges of a graph into edge-disjoint simple paths. Gallai conjectured that every connected $n$-vertex graph has a path cover with at most $\lceil n/2 \rceil$ paths. We prove Gallai's conjecture for…

In a book embedding of a graph G, the vertices of G are placed in order along a straight-line called spine of the book, and the edges of G are drawn on a set of half-planes, called the pages of the book, such that two edges drawn on a page…

We construct a graph G such that any embedding of G into R^{3} contains a nonsplit link of two components, where at least one of the components is a nontrivial knot. Further, for any m < n we produce a graph H so that every embedding of H…

In this paper, for each graph G, a free edge set F is defined. To study the existence of cycle double cover, the naive cycle double cover of G and F have been defined and studied. In the main theorem, the paper, based on the Kuratowski…

We study straight-line drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected…

We prove a new sufficient condition for a cubic 3-connected planar graph to be Hamiltonian. This condition is most easily described as a property of the dual graph. Let $G$ be a planar triangulation. Then the dual $G^*$ is a cubic…

We approach the cycle double cover conjecture by looking for a circular 2-cell embedding of cubic graphs on an arbitrary surface. It is easy to see that if such an embedding exists, we can get to it from an arbitrary starting 2-cell…

A zigzag in a map (a $2$-cell embedding of a connected graph in a connected closed $2$-dimensional surface) is a cyclic sequence of edges satisfying the following conditions: 1) any two consecutive edges lie on the same face and have a…

Consider a graph with a rotation system, namely, for every vertex, a circular ordering of the incident edges. Given such a graph, an angle cover maps every vertex to a pair of consecutive edges in the ordering -- an angle -- such that each…

A universal representation theorem is derived that shows any graph is the intersection graph of one chordal graph, a number of co-bipartite graphs, and one unit interval graph. Central to the the result is the notion of the clique cover…

We classify the planar cubic Cayley graphs of connectivity 2, providing an explicit presentation and embedding for each of them. Combined with [9] this yields a complete description of all planar cubic Cayley graphs.